1st time to Seoul Korea

Go BUsan

Will you only be staying within Seoul? If you want to experience real, authentic Korea I would suggest Insadong - it feels a lot more traditional and a little less touristy. There are a lot of traditional crafts and some really delicious restaurants that serve really authentic, home-cooked style food. I also suggest visiting some of the tea houses in that neighborhood.

If you'd like to go to a real market out door area, I suggest Namdaemun Market! You can grab some really delicious street food there and even some cheap gifts/souvenirs.

I agree with Saeyou, Insadong is a great place to go. It's actually kinda touristy but also more traditional too. There's almost no English on the signs (even Starbucks is written in Korean) and it's a great place to get souvenirs. Dongdaemun and Namdaemun are also good places for markets.

Besides those there are a ton of smaller market streets around the city! It depends on what area you'll be in! I also like shopping at the Gangnam underground subway. It's not food, but they have tons of accessories.